Effects of measurement error on Monte Carlo integration estimators of tree volume: critical height sampling and vertical Monte Carlo methods

The effects of measurement error on Monte Carlo (MC) integration estimators of individual-tree volume that sample upper-stem heights at randomly selected cross-sectional areas (termed vertical methods) were studied. These methods included critical height sampling (on an individual-tree basis), vertical importance sampling (VIS), and vertical control variate sampling (VCS). These estimators were unbiased in the presence of two error models: additive measurement error with mean zero and multiplicative measurement error with mean one. Exact mathematical expressions were derived for the variances of VIS and VCS that include additive components for sampling error and measurement error, which together comprise total variance. Previous studies of sampling error for MC integration estimators of tree volume were combined with estimates of upper-stem measurement error obtained from the mensurational literature to compute typical estimates of total standard errors for VIS and VCS. Through examples, it is shown that measurement error can substantially increase the total root mean square error of the volume estimate, especially for small trees.

Models for Measured Income

Measurement error can impact estimator precision, obscure estimated relationships between variables, and distort the estimated intertemporal behavior of important economic characteristics. A commonly known model for measurement error assumes that measured income is the product of true income and a multiplicative measurement error, which is distributed independently of the level of true income. Based on this model, we derive a collection of flexible parametric forms for the distribution of measured income. We feel that this work could serve as an important reference for measurement error modeling.

Some Variations on Standard Income Measurement Error Models

Measurement error can have a significant impact on measures of inequality. Using a fairly flexible parametric specification of an independent multiplicative measurement error (IMME) model we explore the relationship between changes in the variance of measurement error, for a given mean of measurement error, on the Gini Coefficient. While the measured Gini is greater than the true Gini, the difference decreases as the variance of measurement error decreases. Copulas are used to relax the assumption of independence of measurement error and true income. In this case the measured Gini can be larger or smaller than the true Gini, depending on the correlation between true income and measurement error. Using the same approach with simulations the effect of a different distribution of measurement error is investigated.

Measurement Error and the Distribution of Income

Measurement error can impact estimator precision, obscure estimated relationships between variables, and distort the estimated inter-temporal behaviour of important economic characteristics. A model for measurement error is presented, based on the common assumption that measured income is the product of true income and multiplicative measurement error which is distributed independently of the level of true income. Flexible parametric forms are utilized to model the distributions of true income (generalized gamma) and measurement error (inversed generalized gamma). The corresponding probability density of measures income is shown to be a generalized beta of the second kind (GB2) which can be estimated using MLE. An identification problem is solved with additional information as to the average function of true income reported. The procedure is applied to income data from several Latin American economies. It is found, in some cases, that true and measured income inequality move in opposite directions over time. This finding has important implications for the evaluation of policies designed to affect relative equality in the distribution of income and underscores the importance of obtaining accurate data.